In YDSE, the sources is red ligth of wavelength `7 xx 10^(-7) m`. When a thin glass plate of refractive index 1.5 is put in the path of one of the interfering beams, the central bright fringe shifts by `10^(-3)` m to the position previously occupied by the 5th bright fringe.
What is the thickness of the plate?

Options

A. `5 mu m`
B. `0.005 mu m`
C. `7 mu m`
D. `0.007 mu m`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a Young's Double Slit Experiment (YDSE) setup with red light of wavelength \( \lambda = 7 \times 10^{-7} \, \text{m} \). A thin glass plate with a refractive index \( \mu = 1.5 \) is placed in the path of one of the beams, causing a shift in the central bright fringe.

### Step 2: Identify the shift
The problem states that the central bright fringe shifts by \( 10^{-3} \, \text{m} \) (or 1 mm) to the position previously occupied by the 5th bright fringe.

### Step 3: Use the formula for fringe shift
The formula for the shift \( \Delta y \) in YDSE when a glass plate is introduced is given by:
\[ \Delta y = \frac{(\mu - 1) T D}{d} \]
where:
- \( \Delta y \) is the shift,
- \( \mu \) is the refractive index of the glass,
- \( T \) is the thickness of the glass plate,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

### Step 4: Position of the 5th bright fringe
The position of the \( n \)-th bright fringe is given by:
\[ y_n = \frac{n \lambda D}{d} \]
For the 5th bright fringe (\( n = 5 \)):
\[ y_5 = \frac{5 \lambda D}{d} \]

### Step 5: Set up the equation
Since the central maximum shifts to the position of the 5th bright fringe:
\[ \Delta y = y_5 \]
Substituting the expressions we have:
\[ \frac{(\mu - 1) T D}{d} = \frac{5 \lambda D}{d} \]

### Step 6: Simplify the equation
We can cancel \( D \) and \( d \) from both sides:
\[ (\mu - 1) T = 5 \lambda \]

### Step 7: Solve for thickness \( T \)
Now, substituting the values:
- \( \mu = 1.5 \)
- \( \lambda = 7 \times 10^{-7} \, \text{m} \)

We have:
\[ (1.5 - 1) T = 5 \times (7 \times 10^{-7}) \]
\[ 0.5 T = 35 \times 10^{-7} \]
\[ T = \frac{35 \times 10^{-7}}{0.5} = 70 \times 10^{-7} = 7 \times 10^{-6} \, \text{m} \]

### Step 8: Convert to micrometers
Converting to micrometers:
\[ T = 7 \, \mu m \]

### Final Answer
The thickness of the glass plate is \( 7 \, \mu m \).

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